1. Holt McDougal Algebra 2 13-3 The Unit Circle Toolbox p. 947(1-34) 13.3a 13.3b radian degrees unit circle

2. Holt McDougal Algebra 2 13-3 The Unit Circle How do you convert angle measures between degrees and radians? How do you find the values of trigonometric functions on the unit circle? Essential Questions

3. Holt McDougal Algebra 2 13-3 The Unit Circle So far, you have measured angles in degrees. You can also measure angles in radians. A radian is a unit of angle measure based on arc length. Recall from geometry that an arc is an unbroken part of a circle. If a central angle θ in a circle of radius r, then the measure of θ is defined as 1 radian.

4. Holt McDougal Algebra 2 13-3 The Unit Circle The circumference of a circle of radius r is 2r. Therefore, an angle representing one complete clockwise rotation measures 2 radians. You can use the fact that 2 radians is equivalent to 360° to convert between radians and degrees.

5. Holt McDougal Algebra 2 13-3 The Unit Circle or

6. Holt McDougal Algebra 2 13-3 The Unit Circle Example 1: Converting Between Degrees and Radians Convert each measure from degrees to radians or from radians to degrees. A. – 60° B..

7. Holt McDougal Algebra 2 13-3 The Unit Circle Angles measured in radians are often not labeled with the unit. If an angle measure does not have a degree symbol, you can usually assume that the angle is measured in radians. Reading Math

8. Holt McDougal Algebra 2 13-3 The Unit Circle A unit circle is a circle with a radius of 1 unit. For every point P(x, y) on the unit circle, the value of r is 1. Therefore, for an angle θ in the standard position:

9. Holt McDougal Algebra 2 13-3 The Unit Circle So the coordinates of P can be written as (cosθ, sinθ). The diagram shows the equivalent degree and radian measure of special angles, as well as the corresponding x- and y-coordinates of points on the unit circle.

10. Holt McDougal Algebra 2 13-3 The Unit Circle Example 2A: Using the Unit Circle to Evaluate Trigonometric Functions Use the unit circle to find the exact value of each trigonometric function. cos 225° The angle passes through the point on the unit circle. cos 225° = x Use cos θ = x.

11. Holt McDougal Algebra 2 13-3 The Unit Circle tan Example 2B: Using the Unit Circle to Evaluate Trigonometric Functions Use the unit circle to find the exact value of each trigonometric function. The angle passes through the point on the unit circle. Use tan θ =.

12. Holt McDougal Algebra 2 13-3 The Unit Circle You can use reference angles and Quadrant I of the unit circle to determine the values of trigonometric functions. Trigonometric Functions and Reference Angles

13. Holt McDougal Algebra 2 13-3 The Unit Circle The diagram shows how the signs of the trigonometric functions depend on the quadrant containing the terminal side of θ in standard position.

14. Holt McDougal Algebra 2 13-3 The Unit Circle Example 3: Using Reference Angles to Evaluate Trigonometric functions Use a reference angle to find the exact value of the sine, cosine, and tangent of 330°. Step 1 Find the measure of the reference angle. The reference angle measures 30°

15. Holt McDougal Algebra 2 13-3 The Unit Circle Example 3 Continued Step 2 Find the sine, cosine, and tangent of the reference angle. Use sin θ = y. Use cos θ = x.

16. Holt McDougal Algebra 2 13-3 The Unit Circle Example 3 Continued Step 3 Adjust the signs, if needed. In Quadrant IV, sin θ is negative. In Quadrant IV, cos θ is positive. In Quadrant IV, tan θ is negative.